1st & 2nd Order Spatial Point Patterns Analysis Methods

Analysis
R
sf
tmap
tidyverse
raster
spatstat
Author

Brian Lim

Published

August 29, 2024

Modified

August 29, 2024

3.1 Exercise Overview

In this hands-on exercise, I learn how to to analyze spatial point patterns in R. On top of that I learn to apply first- and second-order analyses to assess the randomness of point distributions, and how to visualize and interpret the spatial concentration of facilities using Kernel Density Estimation (KDE)

Spatial Point Pattern Analysis involves evaluating the distribution of a set of points on a surface. These points can represent the locations of various events or facilities, such as:

  • Events: Crime occurrences, traffic accidents, or disease outbreaks.
  • Facilities: Business services like coffee shops, fast food outlets, or essential facilities such as childcare and eldercare centers.

In this hands-on exercise, we aim to explore the spatial distribution of childcare centers in Singapore using functions from the spatstat package. Specifically, we will address the following questions:

  • Randomness of Distribution: Are childcare centers in Singapore randomly distributed across the country?
  • Clusters and Concentrations: If the distribution is not random, where are the areas with higher concentrations of childcare centers?

3.2 Data Acquisition

Three data set will be used to answer these question. They are:

  1. CHILDCARE: A point feature dataset containing the location and attribute information of childcare centers in Singapore. This dataset was downloaded from Data.gov.sg and is in GeoJSON format.
  2. MP14_SUBZONE_WEB_PL: A polygon feature dataset providing information on URA’s 2014 Master Plan Planning Subzone boundaries. This data is in ESRI Shapefile format and was also downloaded from Data.gov.sg.
  3. CostalOutline: A polygon feature dataset showing the national boundary of Singapore. This dataset is provided by the Singapore Land Authority (SLA) and is in ESRI Shapefile format.

3.3 Getting Started

For this exercise, we will use the following 5 R packages:

  • sf, a relatively new R package specially designed to import, manage and process vector-based geospatial data in R.

  • spatstat, a comprehensive package for point pattern analysis. We’ll use it to perform first- and second-order spatial point pattern analyses and to derive kernel density estimation (KDE) layers.

  • raster, a package for reading, writing, manipulating, and modeling gridded spatial data (rasters). We will use it to convert image outputs generated by spatstat into raster format.

  • maptools, a set of tools for manipulating geographic data, mainly used here to convert spatial objects into the ppp format required by spatstat.

  • tmap, a package for creating high-quality static and interactive maps, leveraging the Leaflet API for interactive visualizations.

To install and load these packages into the R environment, we use the p_load function from the pacman package:

pacman::p_load(sf, raster, spatstat, tmap, tidyverse)

3.4 Importing Data into R

3.4.1 Importing the spatial data

In this section, we’ll use the st_read() function from the sf package to import three geospatial datasets into R:

childcare_sf <- st_read("data/child-care-services-geojson.geojson") %>%
  st_transform(crs = 3414)
Reading layer `child-care-services-geojson' from data source 
  `C:\Users\blzll\OneDrive\Desktop\Y3S1\IS415\Quarto\IS415\Hands-on_Ex\data\child-care-services-geojson.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84
sg_sf <- st_read(dsn = "data", layer="CostalOutline")
Reading layer `CostalOutline' from data source 
  `C:\Users\blzll\OneDrive\Desktop\Y3S1\IS415\Quarto\IS415\Hands-on_Ex\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21
mpsz_sf <- st_read(dsn = "data", 
                layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `C:\Users\blzll\OneDrive\Desktop\Y3S1\IS415\Quarto\IS415\Hands-on_Ex\data' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

The following code chunk changes the referencing system of the newly created simple feature data frames to Singapore national projected coordinate system.

childcare_sf <- st_transform(childcare_sf, crs = 3414)
mpsz_sf <- st_transform(mpsz_sf, crs = 3414)

3.4.2 Mapping the geospatial data sets

After verifying that all datasets share the same CRS, it’s useful to visualize them to confirm their spatial alignment.

tmap_mode("plot")
tm_shape(mpsz_sf) +
  tm_polygons() + 
  tm_shape(childcare_sf) + 
  tm_dots() 

In this code chunk:

  • tm_shape(mpsz_sf): Sets the base layer to the Singapore subzones.
  • tm_polygons(): Plots the subzones with a light blue fill and a dark blue border.
  • tm_shape(childcare_sf): Adds an overlay layer for the childcare centers.
  • tm_dots(): Plots the childcare centers as red dots with a black border.
tmap_mode('view')

tm_shape(childcare_sf) +
  tm_dots()
tmap_mode('plot')

By ensuring that all geospatial layers align correctly within the same map extent, we confirm that their CRS and coordinate values are consistent—a critical aspect of any geospatial analysis.

tmap_mode('plot')

Notice that at the interactive mode, tmap is using leaflet for R API. The advantage of this interactive pin map is it allows us to navigate and zoom around the map freely. We can also query the information of each simple feature (i.e. the point) by clicking of them. Last but not least, you can also change the background of the internet map layer. Currently, three internet map layers are provided. They are: ESRI.WorldGrayCanvas, OpenStreetMap, and ESRI.WorldTopoMap. The default is ESRI.WorldGrayCanvas.

3.5 Geospatial Data wrangling

Although simple feature data frame is gaining popularity again sp’s Spatial* classes, there are, however, many geospatial analysis packages require the input geospatial data in sp’s Spatial* classes. In this section, you will learn how to convert simple feature data frame to sp’s Spatial* class.

3.5.1 Converting sf data frames to sp’s Spatial* class

To work with certain geospatial analysis packages that require data in sp’s Spatial* class, you can convert simple feature (sf) data frames to these classes using the as_Spatial() function from the sf package. Below is the code to convert the sf objects to sp’s Spatial* classes:

childcare <- as_Spatial(childcare_sf)
mpsz <- as_Spatial(mpsz_sf)
sg <- as_Spatial(sg_sf)

After conversion, you can display the information of these three Spatial* classes as follows:

childcare
class       : SpatialPointsDataFrame 
features    : 1545 
extent      : 11203.01, 45404.24, 25667.6, 49300.88  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 2
names       :    Name,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Description 
min values  :   kml_1, <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>018989</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>1, MARINA BOULEVARD, #B1 - 01, ONE MARINA BOULEVARD, SINGAPORE 018989</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>THE LITTLE SKOOL-HOUSE INTERNATIONAL PTE. LTD.</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>08F73931F4A691F4</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 
max values  : kml_999,                  <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>829646</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>200, PONGGOL SEVENTEENTH AVENUE, SINGAPORE 829646</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td>Child Care Services</td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>RAFFLES KIDZ @ PUNGGOL PTE LTD</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>379D017BF244B0FA</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 
mpsz
class       : SpatialPolygonsDataFrame 
features    : 323 
extent      : 2667.538, 56396.44, 15748.72, 50256.33  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 15
names       : OBJECTID, SUBZONE_NO, SUBZONE_N, SUBZONE_C, CA_IND, PLN_AREA_N, PLN_AREA_C,       REGION_N, REGION_C,          INC_CRC, FMEL_UPD_D,     X_ADDR,     Y_ADDR,    SHAPE_Leng,    SHAPE_Area 
min values  :        1,          1, ADMIRALTY,    AMSZ01,      N, ANG MO KIO,         AM, CENTRAL REGION,       CR, 00F5E30B5C9B7AD8,      16409,  5092.8949,  19579.069, 871.554887798, 39437.9352703 
max values  :      323,         17,    YUNNAN,    YSSZ09,      Y,     YISHUN,         YS,    WEST REGION,       WR, FFCCF172717C2EAF,      16409, 50424.7923, 49552.7904, 68083.9364708,  69748298.792 
sg
class       : SpatialPolygonsDataFrame 
features    : 60 
extent      : 2663.926, 56047.79, 16357.98, 50244.03  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +datum=WGS84 +units=m +no_defs 
variables   : 4
names       : GDO_GID, MSLINK, MAPID,              COSTAL_NAM 
min values  :       1,      1,     0,             ISLAND LINK 
max values  :      60,     67,     0, SINGAPORE - MAIN ISLAND 

3.5.2 Converting the Spatial* class into generic sp format

To prepare data for use in the spatstat package, you first need to convert the Spatial* classes into a generic sp format:

childcare_sp <- as(childcare, "SpatialPoints")
sg_sp <- as(sg, "SpatialPolygons")

Then, you can display the properties of these sp objects:

childcare_sp
class       : SpatialPoints 
features    : 1545 
extent      : 11203.01, 45404.24, 25667.6, 49300.88  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
sg_sp
class       : SpatialPolygons 
features    : 60 
extent      : 2663.926, 56047.79, 16357.98, 50244.03  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +datum=WGS84 +units=m +no_defs 

3.5.3 Converting the generic sp format into spatstat’s ppp format

To analyze spatial point patterns, you need to convert the sp objects into ppp objects, which are used by the spatstat package:

childcare_ppp <- as.ppp(st_coordinates(childcare_sf), st_bbox(childcare_sf))
childcare_ppp
Marked planar point pattern: 1545 points
marks are numeric, of storage type  'double'
window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units

Now, let us plot childcare_ppp and examine the different.

plot(childcare_ppp)

You can take a quick look at the summary statistics of the newly created ppp object by using the code chunk below.

summary(childcare_ppp)
Marked planar point pattern:  1545 points
Average intensity 1.91145e-06 points per square unit

*Pattern contains duplicated points*

Coordinates are given to 11 decimal places

marks are numeric, of type 'double'
Summary:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
      0       0       0       0       0       0 

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units

Notice the warning message about duplicates. In spatial point patterns analysis an issue of significant is the presence of duplicates. The statistical methodology used for spatial point patterns processes is based largely on the assumption that process are simple, that is, that the points cannot be coincident.

3.5.4 Handling duplicated points

To determine the existence of duplicate points, we can check for duplicates in your Point Pattern (PPP) object object by using the code chunk below.

any(duplicated(childcare_ppp))
[1] TRUE

To count the number of points at each location, we can use the multiplicity() function as shown in the code chunk below.

multiplicity(childcare_ppp)
   1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16 
   1    1    1    3    1    1    1    1    2    1    1    1    1    1    1    1 
  17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32 
   1    1    1    1    1    1    1    1    1    1    9    1    1    1    1    1 
  33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
  49   50   51   52   53   54   55   56   57   58   59   60   61   62   63   64 
   1    1    1    1    1    1    2    1    1    3    1    1    1    1    1    1 
  65   66   67   68   69   70   71   72   73   74   75   76   77   78   79   80 
   1    1    1    1    1    2    1    1    1    1    1    2    1    1    1    1 
  81   82   83   84   85   86   87   88   89   90   91   92   93   94   95   96 
   1    1    1    3    1    1    1    1    1    1    1    1    1    1    1    1 
  97   98   99  100  101  102  103  104  105  106  107  108  109  110  111  112 
   1    1    1    1    1    1    1    1    2    1    1    1    1    1    1    1 
 113  114  115  116  117  118  119  120  121  122  123  124  125  126  127  128 
   1    1    1    1    1    1    2    1    1    1    3    1    1    1    2    1 
 129  130  131  132  133  134  135  136  137  138  139  140  141  142  143  144 
   1    1    1    1    1    2    1    1    1    1    1    1    1    1    3    2 
 145  146  147  148  149  150  151  152  153  154  155  156  157  158  159  160 
   1    2    1    1    1    2    2    3    1    5    1    5    1    1    1    2 
 161  162  163  164  165  166  167  168  169  170  171  172  173  174  175  176 
   1    1    1    1    2    1    1    1    1    1    1    2    1    1    1    1 
 177  178  179  180  181  182  183  184  185  186  187  188  189  190  191  192 
   1    4    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 193  194  195  196  197  198  199  200  201  202  203  204  205  206  207  208 
   1    1    1    1    1    2    2    1    1    1    1    2    1    4    1    1 
 209  210  211  212  213  214  215  216  217  218  219  220  221  222  223  224 
   2    1    1    1    1    1    1    1    1    1    1    1    2    1    1    1 
 225  226  227  228  229  230  231  232  233  234  235  236  237  238  239  240 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 241  242  243  244  245  246  247  248  249  250  251  252  253  254  255  256 
   1    1    1    1    2    1    1    1    1    1    1    1    1    1    1    1 
 257  258  259  260  261  262  263  264  265  266  267  268  269  270  271  272 
   1    1    1    1    1    1    1    1    1    1    2    1    1    1    1    3 
 273  274  275  276  277  278  279  280  281  282  283  284  285  286  287  288 
   1    1    1    1    1    1    3    1    1    1    1    1    1    1    1    1 
 289  290  291  292  293  294  295  296  297  298  299  300  301  302  303  304 
   1    1    1    1    1    1    1    9    1    1    2    1    1    1    1    1 
 305  306  307  308  309  310  311  312  313  314  315  316  317  318  319  320 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 321  322  323  324  325  326  327  328  329  330  331  332  333  334  335  336 
   1    1    1    5    1    1    1    1    1    2    1    1    2    2    1    1 
 337  338  339  340  341  342  343  344  345  346  347  348  349  350  351  352 
   1    1    1    1    1    1    1    1    1    1    1    1    1    2    2    1 
 353  354  355  356  357  358  359  360  361  362  363  364  365  366  367  368 
   1    1    1    1    9    1    1    1    1    1    1    1    1    1    1    1 
 369  370  371  372  373  374  375  376  377  378  379  380  381  382  383  384 
   1    3    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 385  386  387  388  389  390  391  392  393  394  395  396  397  398  399  400 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 401  402  403  404  405  406  407  408  409  410  411  412  413  414  415  416 
   1    1    2    1    1    1    1    1    1    1    2    1    1    1    1    1 
 417  418  419  420  421  422  423  424  425  426  427  428  429  430  431  432 
   1    1    1    1    1    1    1    2    1    1    2    1    1    1    1    1 
 433  434  435  436  437  438  439  440  441  442  443  444  445  446  447  448 
   1    1    1    1    2    1    1    1    1    1    1    1    1    1    1    1 
 449  450  451  452  453  454  455  456  457  458  459  460  461  462  463  464 
   1    1    9    9    1    1    1    1    1    1    1    1    1    1    2    1 
 465  466  467  468  469  470  471  472  473  474  475  476  477  478  479  480 
   2    1    1    1    1    1    1    1    1    1    1    1    2    2    1    1 
 481  482  483  484  485  486  487  488  489  490  491  492  493  494  495  496 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 497  498  499  500  501  502  503  504  505  506  507  508  509  510  511  512 
   1    1    1    1    1    1    2    1    1    1    1    1    1    1    1    2 
 513  514  515  516  517  518  519  520  521  522  523  524  525  526  527  528 
   1    1    1    1    1    1    1    1    1    1    1    2    1    1    3    1 
 529  530  531  532  533  534  535  536  537  538  539  540  541  542  543  544 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 545  546  547  548  549  550  551  552  553  554  555  556  557  558  559  560 
   1    1    1    1    1    1    1    1    1    3    1    1    1    1    1    1 
 561  562  563  564  565  566  567  568  569  570  571  572  573  574  575  576 
   2    2    2    1    1    1    1    2    1    1    2    1    1    1    2    1 
 577  578  579  580  581  582  583  584  585  586  587  588  589  590  591  592 
   1    2    1    1    1    1    1    9    1    4    1    2    1    1    1    1 
 593  594  595  596  597  598  599  600  601  602  603  604  605  606  607  608 
   2    1    1    1    1    1    1    1    2    1    2    1    1    1    1    1 
 609  610  611  612  613  614  615  616  617  618  619  620  621  622  623  624 
   1    1    1    1    1    1    1    1    1    2    1    2    1    1    1    1 
 625  626  627  628  629  630  631  632  633  634  635  636  637  638  639  640 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 641  642  643  644  645  646  647  648  649  650  651  652  653  654  655  656 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    4 
 657  658  659  660  661  662  663  664  665  666  667  668  669  670  671  672 
   1    1    1    1    1    1    1    3    1    1    1    1    1    1    1    1 
 673  674  675  676  677  678  679  680  681  682  683  684  685  686  687  688 
   1    1    1    1    1    4    1    1    1    1    1    4    1    1    1    1 
 689  690  691  692  693  694  695  696  697  698  699  700  701  702  703  704 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 705  706  707  708  709  710  711  712  713  714  715  716  717  718  719  720 
   1    1    2    1    1    1    1    1    1    1    1    1    1    1    1    1 
 721  722  723  724  725  726  727  728  729  730  731  732  733  734  735  736 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 737  738  739  740  741  742  743  744  745  746  747  748  749  750  751  752 
   1    2    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 753  754  755  756  757  758  759  760  761  762  763  764  765  766  767  768 
   1    1    1    1    1    2    1    1    1    1    1    1    1    1    1    1 
 769  770  771  772  773  774  775  776  777  778  779  780  781  782  783  784 
   1    1    1    1    1    1    1    1    1    4    1    1    1    1    1    1 
 785  786  787  788  789  790  791  792  793  794  795  796  797  798  799  800 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 801  802  803  804  805  806  807  808  809  810  811  812  813  814  815  816 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 817  818  819  820  821  822  823  824  825  826  827  828  829  830  831  832 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 833  834  835  836  837  838  839  840  841  842  843  844  845  846  847  848 
   1    1    1    1    1    1    1    2    1    1    1    1    1    1    1    1 
 849  850  851  852  853  854  855  856  857  858  859  860  861  862  863  864 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 865  866  867  868  869  870  871  872  873  874  875  876  877  878  879  880 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    2 
 881  882  883  884  885  886  887  888  889  890  891  892  893  894  895  896 
   3    1    1    1    2    1    1    1    3    1    1    3    1    1    1    1 
 897  898  899  900  901  902  903  904  905  906  907  908  909  910  911  912 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 913  914  915  916  917  918  919  920  921  922  923  924  925  926  927  928 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 929  930  931  932  933  934  935  936  937  938  939  940  941  942  943  944 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 945  946  947  948  949  950  951  952  953  954  955  956  957  958  959  960 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    2 
 961  962  963  964  965  966  967  968  969  970  971  972  973  974  975  976 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 977  978  979  980  981  982  983  984  985  986  987  988  989  990  991  992 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 993  994  995  996  997  998  999 1000 1001 1002 1003 1004 1005 1006 1007 1008 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 
   1    1    1    1    1    1    1    1    1    2    2    1    1    1    1    1 
1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 
   1    1    1    1    1    2    1    1    1    1    1    1    1    1    1    1 
1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 
   1    1    1    1    1    1    1    1    2    2    1    1    1    5    1    1 
1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 
   1    1    1    1    1    1    1    1    1    2    1    1    1    1    1    1 
1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 
   1    1    1    1    1    1    1    1    1    1    2    1    1    1    1    1 
1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 
   1    9    1    2    2    1    1    1    2    1    1    1    1    1    1    1 
1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 
   1    1    1    1    2    1    1    1    3    1    1    1    1    1    1    1 
1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 
   9    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 
   1    1    1    2    1    1    1    1    1    1    1    1    1    1    1    1 
1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    2 
1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 
   1    1    1    2    1    2    1    1    1    2    2    2    1    1    1    1 
1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 
   1    1    2    1    1    1    1    1    1    1    1    1    2    1    1    1 
1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 
   1    1    1    1    3    1    1    1    1    1    1    1    1    1    1    1 
1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 
   1    1    1    1    1    1    1    1    4    1    1    1    1    1    2    1 
1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 
   1    1    1    1    1    1    1    1    1    9    1    1    1    1    1    1 
1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    2    1 
1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 
   1    2    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 
   1    1    1    1    1    1    1    1    1    1    2    1    1    1    1    1 
1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 
   1    1    1    1    1    1    2    1    1    1    1    1    1    1    1    1 
1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 
   1    1    1    1    1    1    1    1    1    1    5    1    1    1    1    1 
1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 
   1    1    1    1    1    2    1    1    1    1    2    1    1    1    1    3 
1537 1538 1539 1540 1541 1542 1543 1544 1545 
   1    1    1    1    1    1    2    1    1 

To check for the number of locations that have more than one point event, we can use the code chunk below.

sum(multiplicity(childcare_ppp) > 1)
[1] 128

The output show 128 duplicated point events.

To view the locations of these duplicate point events, we will plot childcare data by using the code chunk below.

tmap_mode('view')
tm_shape(childcare) +
  tm_dots(alpha=0.4, 
          size=0.05)
tmap_mode('plot')
tmap_mode('view')

There are three ways to overcome this problem.

  1. The first and easiest way is to delete the duplicates. But, that will also mean that some useful point events will be lost.

  2. The second solution is use jittering. If duplicates are hard to spot, you can apply a slight jitter to the points’ coordinates. Jittering will slightly displace the points so that overlapping points are separated on the map.

  3. The third solution is to make each point “unique” and then attach the duplicates of the points to the patterns as marks, as attributes of the points. Then you would need analytical techniques that take into account these marks.

The jitter parameter will slightly move each point by a small, random amount. This can help to visually separate points that are in the same space.

tm_shape(childcare) +
  tm_dots(jitter=0.1, alpha=0.4, size=0.05)
childcare_ppp_jit <- rjitter(childcare_ppp, 
                             retry=TRUE, 
                             nsim=1, 
                             drop=TRUE)

After jittering we can check if there are any duplicate point in this geospatial data

any(duplicated(childcare_ppp_jit))
[1] FALSE

3.5.5 Creating owin object

To confine analysis to a geographical area, convert the SpatialPolygon object to an owin object of spatstat:

sg_owin <- as.owin(sg_sf)
  
plot(sg_owin)

Further analysis can be done through the summary() function of Base R:

summary(sg_owin)
Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

3.5.6 Combining point events object and owin object

Finally, you can combine the point events with the polygon feature to create a ppp object confined to the Singapore region

childcareSG_ppp = childcare_ppp[sg_owin]
summary(childcareSG_ppp )
Marked planar point pattern:  1545 points
Average intensity 2.129929e-06 points per square unit

*Pattern contains duplicated points*

Coordinates are given to 11 decimal places

marks are numeric, of type 'double'
Summary:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
      0       0       0       0       0       0 

Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401
plot(childcareSG_ppp)

3.6 First-order Spatial Point Patterns Analysis

In this section, we will learn how to perform first-order Spatial Point Pattern Analysis (SPPA) using the spatstat package. The focus will be on:

  1. Deriving Kernel Density Estimation (KDE) layers for visualizing and exploring the intensity of point processes.

  2. Performing Confirmatory Spatial Point Patterns Analysis using Nearest Neighbour statistics.

3.6.1 Kernel Density Estimation

Kernel Density Estimation (KDE) is a non-parametric way to estimate the intensity (density) of spatial point patterns. It helps in visualizing the spatial distribution of points by smoothing the point pattern to create a continuous surface.

3.6.1.1 Computing kernel density estimation using automatic bandwidth selection method

To compute the KDE for the spatial point pattern of childcare services in Singapore, we’ll use the density() function from the spatstat package. This function allows for various configurations:

  • Automatic Bandwidth Selection: We’ll use  bw.diggle() , a method that selects an optimal bandwidth based on the data. Other methods like  bw.CvL() ,  bw.scott(), or  bw.ppl() can also be used depending on the specific needs of the analysis.

  • Smoothing Kernel: The default kernel used is Gaussian. Other options include “Epanechnikov”, “Quartic”, or “Disc”.

  • Edge Correction: The intensity estimate is corrected for edge effects to reduce bias, following methods described by Jones (1993) and Diggle (2010).

kde_childcareSG_bw <- density(childcareSG_ppp,
                              sigma=bw.diggle,
                              edge=TRUE,
                            kernel="gaussian") 

plot(kde_childcareSG_bw)

In this example, the density values range from 0 to 0.000035, which are quite small. This is because the unit of measurement is in meters, making the density values “number of points per square meter”.

To check the bandwidth used:

bw <- bw.diggle(childcareSG_ppp)
bw
   sigma 
298.4095 

3.6.1.2 Rescalling KDE values

The small density values are due to the measurement unit being in meters. To make the results more interpretable, we can rescale the spatial point pattern from meters to kilometers.

The code chunk below rescale the point pattern to kilometers, recompute the KDE with the rescaled data, and plots the rescaled KDE

childcareSG_ppp.km <- rescale.ppp(childcareSG_ppp, 1000, "km")

kde_childcareSG_bw <- density(childcareSG_ppp.km, 
                              sigma=bw.diggle, 
                              edge=TRUE, 
                              kernel="gaussian")
plot(kde_childcareSG_bw)

The KDE output will look identical to the original, but the density values will now be more comprehensible, reflecting “number of points per square kilometer.”

3.6.2 Working with different automatic badwidth methods

Different bandwidth selection methods can produce different smoothing results:

  • bw.CvL(): Cross-validation based on the likelihood.

  • bw.scott(): Scott’s rule of thumb.

  • bw.ppl(): Likelihood cross-validation proposed by Diggle.

bw.CvL(childcareSG_ppp.km)
   sigma 
4.543278 
bw.scott(childcareSG_ppp.km)
 sigma.x  sigma.y 
2.224898 1.450966 
bw.ppl(childcareSG_ppp.km)
    sigma 
0.3897114 
bw.diggle(childcareSG_ppp.km)
    sigma 
0.2984095 

Recommendation: 1. Baddeley et. (2016) suggested the use of the bw.ppl() algorithm because in ther experience it tends to produce the more appropriate values when the pattern consists predominantly of tight clusters.

  1. But they also insist that if the purpose of once study is to detect a single tight cluster in the midst of random noise then the bw.diggle() method seems to work best.
kde_childcareSG.ppl <- density(childcareSG_ppp.km, 
                               sigma=bw.ppl, 
                               edge=TRUE,
                               kernel="gaussian")
childcareSG_ppp.km <- rescale.ppp(childcareSG_ppp, 1000, "km")
kde_childcareSG.bw <- density(childcareSG_ppp.km, 
                              sigma=bw.diggle, 
                              edge=TRUE, 
                              kernel="gaussian")
par(mfrow=c(1,2))
plot(kde_childcareSG.bw, main = "bw.diggle")
plot(kde_childcareSG.ppl, main = "bw.ppl")

3.6.3 Working with different kernel methods

Beyond the Gaussian kernel, three other kernels can be used to compute KDE:

  • Epanechnikov

  • Quartic

  • Disc

par(mfrow=c(2,2), mar=c(1, 1, 1, 1), cex=0.5)
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="gaussian"), 
     main="Gaussian")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="epanechnikov"), 
     main="Epanechnikov")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="quartic"), 
     main="Quartic")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="disc"), 
     main="Disc")

3.7 Fixed and Adaptive KDE

3.7.1 Computing KDE by using fixed bandwidth

Next, you will compute a KDE layer by defining a bandwidth of 600 meter. Notice that in the code chunk below, the sigma value used is 0.6. This is because the unit of measurement of childcareSG_ppp.km object is in kilometer, hence the 600m is 0.6km.

To compute a KDE layer using a fixed bandwidth of 0.6 km, use the following code:

kde_childcareSG_600 <- density(childcareSG_ppp.km, sigma=0.6, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG_600)

This will generate a KDE layer with a consistent bandwidth across the entire study area, useful for uniform spatial point patterns.

3.7.2 Computing KDE by using adaptive bandwidth

Adaptive bandwidth methods are more suitable for spatial point patterns with high variability, such as urban versus rural areas. The adaptive.density() function of spatstat can be used to create a KDE layer that adjusts the bandwidth based on point density:

kde_childcareSG_adaptive <- adaptive.density(childcareSG_ppp.km, method="kernel")
plot(kde_childcareSG_adaptive)

To compare the outputs of fixed and adaptive bandwidth KDE:

par(mfrow=c(1,2))
plot(kde_childcareSG.bw, main = "Fixed Bandwidth")
plot(kde_childcareSG_adaptive, main = "Adaptive Bandwidth")

3.7.3 Converting KDE output into grid object.

For mapping purposes, KDE outputs can be converted into grid and raster formats 1. Converting to Grid Object:

gridded_kde_childcareSG_bw <- as(kde_childcareSG.bw, "SpatialGridDataFrame")
spplot(gridded_kde_childcareSG_bw)

  1. Converting to Raster Object:
kde_childcareSG_bw_raster <- raster(kde_childcareSG.bw)
kde_childcareSG_bw_raster
class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : NA 
source     : memory
names      : layer 
values     : -8.476185e-15, 28.51831  (min, max)
  1. Assigning Projection System: Ensure the CRS (Coordinate Reference System) is assigned correctly:
projection(kde_childcareSG_bw_raster) <- CRS("+init=EPSG:3414")
kde_childcareSG_bw_raster
class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +units=m +no_defs 
source     : memory
names      : layer 
values     : -8.476185e-15, 28.51831  (min, max)
  1. Visualizing the Raster: Finally, use the tmap package to visualize the raster in a cartographic map:
tm_shape(kde_childcareSG_bw_raster) + 
  tm_raster("layer", palette = "viridis") +
  tm_layout(legend.position = c("right", "bottom"), frame = FALSE)

Notice that the raster values are encoded explicitly onto the raster pixel using the values in “v”” field.

3.7.4 Comparing Spatial Point Patterns using KDE

In this section, we will learn to compare KDE of childcare at Ponggol, Tampines, Chua Chu Kang and Jurong West planning regions.

3.7.4.1 Extracting study area

pg <- mpsz_sf %>%
  filter(PLN_AREA_N == "PUNGGOL")
tm <- mpsz_sf %>%
  filter(PLN_AREA_N == "TAMPINES")
ck <- mpsz_sf %>%
  filter(PLN_AREA_N == "CHOA CHU KANG")
jw <- mpsz_sf %>%
  filter(PLN_AREA_N == "JURONG WEST")

Plotting target planning areas 1. Ponggol

par(mfrow=c(2,2))
plot(pg, main = "Ponggol")

  1. Tampines
plot(tm, main = "Tampines")

  1. Choa Chu Kang
plot(ck, main = "Choa Chu Kang")

  1. Jurong West
plot(jw, main = "Jurong West")

3.7.4.2 Creating owin object

Now, we will convert these sf objects into owin objects that is required by spatstat.

The owin objects represent the study areas as window objects, which are necessary for spatial point pattern analysis in spatstat. These are created by converting the sf objects (pg, tm, ck, and jw) representing different regions into owin format:

pg_owin = as.owin(pg)
tm_owin = as.owin(tm)
ck_owin = as.owin(ck)
jw_owin = as.owin(jw)

3.7.4.3 Combining childcare points and the study area

we are then able to extract childcare that is within the specific region to perform analysis later on.

childcare_pg_ppp = childcare_ppp_jit[pg_owin]
childcare_tm_ppp = childcare_ppp_jit[tm_owin]
childcare_ck_ppp = childcare_ppp_jit[ck_owin]
childcare_jw_ppp = childcare_ppp_jit[jw_owin]

The childcare centers within each specific region are extracted using the owin objects.

These point patterns (ppp objects) are then rescaled from meters to kilometers:

childcare_pg_ppp.km = rescale.ppp(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale.ppp(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale.ppp(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale.ppp(childcare_jw_ppp, 1000, "km")

Finally, the four study areas and the locations of the childcare centers are plotted:

par(mfrow=c(2,2), mar=c(1, 1, 1, 1), cex=0.5)
plot(childcare_pg_ppp.km, main="Punggol")
plot(childcare_tm_ppp.km, main="Tampines")
plot(childcare_ck_ppp.km, main="Choa Chu Kang")
plot(childcare_jw_ppp.km, main="Jurong West")

3.7.4.4 Computing KDE

The Kernel Density Estimate (KDE) for each area is computed using the density() function, with the bw.diggle method to derive the bandwidth:

par(mfrow=c(2,2), mar=c(1, 1, 1, 1), cex=0.5)
plot(density(childcare_pg_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian"), main="Punggol")
plot(density(childcare_tm_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian"), main="Tampines")
plot(density(childcare_ck_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian"), main="Choa Chu Kang")
plot(density(childcare_jw_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian"), main="Jurong West")

3.7.4.5 Computing Fixed Bandwidth KDE

For comparison, a fixed bandwidth of 250 meters is used to compute KDE for the same areas:

par(mfrow=c(2,2), mar=c(1, 1, 1, 1), cex=0.5)
plot(density(childcare_pg_ppp.km, sigma=0.25, edge=TRUE, kernel="gaussian"), main="Punggol")
plot(density(childcare_tm_ppp.km, sigma=0.25, edge=TRUE, kernel="gaussian"), main="Tampines")
plot(density(childcare_ck_ppp.km, sigma=0.25, edge=TRUE, kernel="gaussian"), main="Choa Chu Kang")
plot(density(childcare_jw_ppp.km, sigma=0.25, edge=TRUE, kernel="gaussian"), main="Jurong West")

3.8 Nearest Neighbour Analysis

In this section, we will perform the Clark-Evans test of aggregation for a spatial point pattern by using clarkevans.test() of statspat.

3.8.1 Testing Spatial Point Patterns using Clark and Evans Test

The Clark-Evans test is performed to assess the spatial distribution of the childcare centers. The hypotheses are:

  • H0: The distribution of childcare centers is random.

  • H1: The distribution of childcare centers is clustered.

The test is conducted as follows:

clarkevans.test(childcareSG_ppp, correction="none", clipregion="sg_owin", alternative="clustered", nsim=99)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcareSG_ppp
R = 0.55631, p-value < 2.2e-16
alternative hypothesis: clustered (R < 1)

Result Interpretation:

  • R = 0.55631: The R-value is less than 1, indicating a tendency towards clustering.

  • p-value < 2.2e-16: The p-value is extremely small, suggesting that the clustering pattern is statistically significant. The null hypothesis of CSR (Complete Spatial Randomness) is rejected in favor of the alternative hypothesis, indicating that the childcare centers in Singapore are clustered.

3.8.2 Clark and Evans Test: Punggol planning area

In the code chunk below,  clarkevans.test() of spatstat is used to performs Clark-Evans test of aggregation for childcare centre in Punggol planning area.

clarkevans.test(childcare_pg_ppp,
                correction="none",
                clipregion=NULL,
                alternative="two.sided",
                nsim=999)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcare_pg_ppp
R = 0.95031, p-value = 0.4578
alternative hypothesis: two-sided

Interpretation:

  1. R = 0.91163: The R-value is close to 1, indicating a spatial distribution that is close to random.
  • p-value = 0.1867: The p-value is greater than 0.05, meaning there is no statistically significant evidence to reject the null hypothesis of CSR. This suggests that the childcare centers in the Punggol area are randomly distributed and do not exhibit significant clustering or regularity.

3.9 Second-order Spatial Point Patterns Analysis

This section introduces second-order analyses of spatial point patterns, focusing on measuring interaction between points.

3.10 Analysing Spatial Point Process Using G-Function

The G function measures the distribution of the distances from an arbitrary event to its nearest event. In this section, you will learn how to compute G-function estimation by using Gest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

3.10.1 Choa Chu Kang planning area

3.10.1.1 Computing G-function estimation

The code chunk below is used to compute G-function using Gest() of spatat package.

G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK, xlim=c(0,500))

3.10.1.2 Performing Complete Spatial Randomness Test

To perform a Complete Spatial Randomness (CSR) test using a Monte Carlo simulation with the G-function in R, you are correctly using the envelope() function from the spatstat package. Here’s how you can carry out the test and interpret the results:

  1. Hypothesis Definition:

    • Null Hypothesis (Ho): The distribution of childcare services at Choa Chu Kang is randomly distributed (i.e., follows CSR).

    • Alternative Hypothesis (H1): The distribution of childcare services at Choa Chu Kang is not randomly distributed.

  2. Set Up the Test:

    • You will perform a Monte Carlo test using the G-function, which measures the distribution of nearest-neighbor distances.
G_CK.csr <- envelope(childcare_ck_ppp, Gest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_CK.csr)

3.10.2 Tampines planning area

3.10.2.1 Computing G-function estimation

G_tm = Gest(childcare_tm_ppp, correction = "best")
plot(G_tm)

3.10.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, we perform the Complete Spatial Randomness (CSR) test for the distribution of childcare services in Tampines using the G-function in R, you can follow the steps and code chunk below:

  • Hypothesis:

    • Null Hypothesis (Ho): The distribution of childcare services at Tampines is randomly distributed (CSR).

    • Alternative Hypothesis (H1): The distribution of childcare services at Tampines is not randomly distributed

G_tm.csr <- envelope(childcare_tm_ppp, Gest, correction = "all", nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_tm.csr)

3.11 Analysing Spatial Point Process Using F-Function

The F-function estimates the empty space function F(r) from a point pattern within a defined window. It provides insights into the spatial distribution by measuring the distribution of distances from a randomly chosen location in the study area to the nearest event (e.g., childcare centers). We will compute the F-function for the Choa Chu Kang and Tampines planning areas and perform a Complete Spatial Randomness (CSR) test using Monte Carlo simulations.

We will learn how to compute F-function estimation by using Fest() of spatstat package, and how to perform monta carlo simulation test using envelope() of spatstat package.

3.11.1 Choa Chu Kang planning area

3.11.1.1 Computing F-fucntion estimate

F_CK = Fest(childcare_ck_ppp)
plot(F_CK)

3.11.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns at Choa Chu Kang, a hypothesis test using the F-function will be conducted. The hypotheses are:

  • Ho (Null Hypothesis): The distribution of childcare services at Choa Chu Kang is randomly distributed.

  • H1 (Alternative Hypothesis): The distribution of childcare services at Choa Chu Kang is not randomly distributed.

The null hypothesis will be rejected if the p-value is smaller than the alpha value of 0.001.

Monte Carlo Test Using F-function:

F_CK.csr <- envelope(childcare_ck_ppp, Fest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_CK.csr)

3.11.2 Tampines planning area

3.11.2.1 Computing F-fucntion estimation

Monte Carlo test with F-fucntion

F_tm = Fest(childcare_tm_ppp, correction = "best")
plot(F_tm)

3.11.2.2 Computing F-fucntion estimation

Similar to before, a hypothesis test will be conducted to confirm the observed spatial patterns above. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

F_tm.csr <- envelope(childcare_tm_ppp, Fest, correction = "all", nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_tm.csr)

3.12 Analysing Spatial Point Process Using K-Function

K-function measures the number of events found up to a given distance of any particular event. In this section, you will learn how to compute K-function estimates by using Kest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

3.12.1 Choa Chu Kang planning area

3.12.1.1 Computing K-fucntion estimation

K_ck = Kest(childcare_ck_ppp, correction = "Ripley")
plot(K_ck, . -r ~ r, ylab= "K(d)-r", xlab = "d(m)")

3.12.1.2 Performing Complete Spatial Randomness Test

Similar to before, a hypothesis test will be conducted to confirm the observed spatial patterns above. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")

3.12.2 Tampines planning area

3.12.2.1 Computing K-fucntion estimation

K_tm = Kest(childcare_tm_ppp, correction = "Ripley")
plot(K_tm, . -r ~ r, 
     ylab= "K(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

3.12.2.2 Performing Complete Spatial Randomness Test

Similar to before, a hypothesis test will be conducted to confirm the observed spatial patterns above. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_tm.csr, . - r ~ r, 
     xlab="d", ylab="K(d)-r", xlim=c(0,500))

3.13 Analysing Spatial Point Process Using L-Function

The L-function is a method used to analyze spatial point patterns by transforming the K-function to be more interpretable, particularly by normalizing the function against a theoretical CSR model. This section covers how to compute the L-function estimation and perform a Monte Carlo simulation test using the  Lest() and envelope() of spatstat package.

3.13.1 Choa Chu Kang planning area

3.13.1.1 Computing L-fucntion estimation

To compute the L-function estimation for Choa Chu Kang, use the Lest() function with the “Ripley” correction. Then, plot the results.

L_ck = Lest(childcare_ck_ppp, correction = "Ripley")
plot(L_ck, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)")

3.13.1.2 Performing Complete Spatial Randomness Test

Similar to before, a hypothesis test will be conducted to confirm the observed spatial patterns above. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")

3.13.2 Tampines planning area

3.13.2.1 Computing L-fucntion estimation

L_tm = Lest(childcare_tm_ppp, correction = "Ripley")
plot(L_tm, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

3.13.2.2 Performing Complete Spatial Randomness Test

Similar to before, a hypothesis test will be conducted to confirm the observed spatial patterns above. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_tm.csr, . - r ~ r, xlab="d", ylab="L(d)-r", xlim=c(0,500))